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Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. (English) Zbl 0762.49017
The paper is concerned with the optimal design problem \[ \inf_ A \int_ \Omega j\bigl(x,u_ A(x)\bigr) dx, \] where \(\Omega{}\) is a given domain of \({\mathbf R}^ n\) (\(n\geq{} 2\)), \(A\) varies in the open subsets of \(\Omega{}\) and \(u_ A\in{} H^ 1_ 0\bigl(A\bigr)\) is the solution of the problem \[ \Delta{} u_ A=-f\quad\quad\hbox{\mathrm in }A. \] The cost functions \(j\) and \(f\in{} L^ 2\bigl(\Omega{}\bigr)\) are given. In general, the infimum is not attained. By using the approximation results of G. Dal Maso and U. Mosco [Appl. Math. Optim. 15, 15-63 (1987; Zbl 0644.35033)], the authors are able to find a relaxed formulation of the problem which always admits a solution. Moreover, they find a set of necessary conditions for optimality. Some of these were already known in the literature [see for instance O. Pironneau, “Optimal shape design for elliptic systems” (1984; Zbl 0534.49001)].
Reviewer: L.Ambrosio (Roma)

MSC:
49Q10 Optimization of shapes other than minimal surfaces
49K15 Optimality conditions for problems involving ordinary differential equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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